Mathematical Researches

fa میانگین پذیری دوری حاصل ضرب لائو و گسترش مدولی جبرهای باناخ Cyclic amenability of Lau product and module extension Banach algebras ریاضی Mat علمی پژوهشی بنیادی S به تازه‌گی نتایجی در مورد میانگین‌پذیری دوری (تقریبی) حاصل‌ضرب لائوی دو جبر باناخ بدست آمده است. در این مقاله ضمن مشخص کردن ضابطه اشتقاق‌های دوری روی حاصل‌ضرب لائوی جبرهای باناخ و گسترش مدولی یک جبر باناخ شرط لازم و کافی برای میانگین‌پذیری دوری (تقریبی) آن‌ها را ارایه می‌نماییم. این نه تنها نتایج تازه‌ای را در مورد میانگین‌پذیری دوری (تقریبی) این دسته از جبرهای باناخ ارایه می‌کند بلکه برخی قضایای اساسی در این خصوص را نیز بهبود می‌بخشد.    Introduction The notion of weak amenability for commutative Banach algebras was introduced and studied for the first time by Bade, Curtis and Dales. Johnson extended this concept to the non commutative case and showed that group algebras of all locally compact groups are weakly amenable. A Banach algebra <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > is called weakly amenable if every continuous derivation <m:omath><m:r>D</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>:</m:r><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>→</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>A</m:r></m:e><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image004.gif" style="width:50px; height:15px" > is inner. It is often useful to restrict one's attention to derivations <m:omath><m:r>D</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>:</m:r><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>→</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>A</m:r></m:e><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image004.gif" style="width:50px; height:15px" > satisfying the property <m:omath><m:r>D</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>a</m:r></m:e></m:d><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>c</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>+</m:r><m:r>D</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>c</m:r></m:e></m:d><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>a</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image006.gif" style="width:119px; height:15px" > for all <m:omath><m:r>a</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>c</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>.</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image008.gif" style="width:42px; height:15px" > Such derivations are called cyclic. Clearly inner derivations are cyclic. A Banach algebra is called cyclic amenable if every continuous cyclic derivations <m:omath><m:r>D</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>:</m:r><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>→</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>A</m:r></m:e><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image004.gif" style="width:50px; height:15px" >is inner. This notion was presented by Gronbaek. He investigated the hereditary properties of this concept, found some relations between cyclic amenability of a Banach algebra and the trace extension property of its ideals. Ghahramani and Loy introduced several approximate notions of amenability by requiring that all bounded derivations from a given Banach algebra <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > into certain Banach <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" >-bimodules to be approximately inner. In the same paper and the subsequent one, the authors showed the distinction between each of these concepts and the corresponding classical notions and investigated properties of algebras in each of these new classes. Motivated by this notions, Esslamzadeh and Shojaee defined the concept of approximate cyclic amenability for Banach algebras and investigated the hereditary properties for this new notion. Periliminaries Let <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > be a Banach algebra and let <m:omath><m:r>X</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image010.gif" style="width:8px; height:15px" > be an <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" >-bimodule. Then the <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>l</m:r></m:e><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image012.gif" style="width:11px; height:15px" >-direct sum <m:omath><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>×</m:r><m:r>X</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image014.gif" style="width:30px; height:15px" > under the multiplication <m:omathpara><m:omath><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>a</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>x</m:r></m:e></m:d><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>b</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>y</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>ab</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>ay</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>+</m:r><m:r>xb</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>              (</m:r><m:r>a</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>b</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>x</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>y</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r>X</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>),</m:r></m:omath></m:omathpara><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image016.gif" style="width:275px; height:15px" > is a Banach algebra called the module extension of <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > by <m:omath><m:r>X</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image010.gif" style="width:8px; height:15px" > and denoted by <m:omath><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>&oplus;</m:r><m:r>X</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:33px; height:15px" >. The class of module extension Banach algebras contains a wide variety of Banach algebras includes a triangular Banach algebra <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Tri(</m:r><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>X</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>B</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>).</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image020.gif" style="width:61px; height:15px" > Every triangular Banach algebra <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Tri(</m:r><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>X</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>B</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>).</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image020.gif" style="width:61px; height:15px" >can be identified with the module extension Banach algebra <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>×</m:r><m:r>B</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)&oplus;</m:r><m:r>X</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image022.gif" style="width:65px; height:15px" >. On the other hand, for two Banach algebra <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > and <m:omath><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image024.gif" style="width:8px; height:15px" > with <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∆</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>(</m:r><m:r>B</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)≠∅</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image026.gif" style="width:48px; height:15px" > and for <m:omath><m:r>θ</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈∆(</m:r><m:r>B</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>)</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image028.gif" style="width:47px; height:15px" >, the set of all non-zero multiplicative linear functionals on <m:omath><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image024.gif" style="width:8px; height:15px" >, the <m:omath><m:r>θ</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image030.gif" style="width:7px; height:15px" >-Lau product <m:omath><m:r>A</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>×</m:r></m:e><m:sub><m:r>θ</m:r></m:sub></m:ssub><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image032.gif" style="width:36px; height:15px" > is a Banach algebra which is defined as the <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>l</m:r></m:e><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image012.gif" style="width:11px; height:15px" >-direct sum <m:omath><m:r>A</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>×</m:r><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image034.gif" style="width:30px; height:15px" > equipped with the algebra multiplication <m:omathpara><m:omath><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>a</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>b</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sub></m:ssub></m:e></m:d><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>a</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>b</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>a</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sub></m:ssub><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>a</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>+</m:r><m:r>θ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>b</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub></m:e></m:d><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>a</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>+</m:r><m:r>θ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>b</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sub></m:ssub></m:e></m:d><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>a</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>b</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sub></m:ssub><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>b</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>                  </m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>a</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>a</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r>A</m:r><m:r>,</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>b</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>b</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:r>B</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>.</m:r></m:omath></m:omathpara> <img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image036.gif" style="width:427px; height:15px" >This type of product was introduced by Lau for certain class of Banach algebras known as Lau algebras and was extended by Sangani Monfared for arbitrary Banach algebras. The unitization <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>A</m:r></m:e><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>♯</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image038.gif" style="width:12px; height:16px" > of <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > can be regarded as the <m:omath><m:r>ι</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image040.gif" style="width:3px; height:15px" >-Lau product <m:omath><m:r>A</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>×</m:r></m:e><m:sub><m:r>ι</m:r></m:sub></m:ssub><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image042.gif" style="width:32px; height:15px" >, where <m:omath><m:r>ι</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈Δ(</m:r><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="p"></m:sty></m:scr></m:rpr>C)</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image044.gif" style="width:42px; height:15px" > is the identity map. This product provides not only new examples of Banach algebras by themselves, but it can also serve as a source of (counter) examples for various purposes in functional and harmonic analysis. From the homological algebra point of view <m:omath><m:r>A</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>×</m:r></m:e><m:sub><m:r>θ</m:r></m:sub></m:ssub><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image032.gif" style="width:36px; height:15px" > is a strongly splitting Banach algebra extension of <m:omath><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image024.gif" style="width:8px; height:15px" > by <m:omath><m:r>A</m:r><m:r>.</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image046.gif" style="width:10px; height:15px" > The Lau product of Banach algebras enjoys some properties that are not shared in general by arbitrary strongly splitting extensions. For instance, commutativity is not preserved by a generally strongly splitting extension. However, <m:omath><m:r>A</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>×</m:r></m:e><m:sub><m:r>θ</m:r></m:sub></m:ssub><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image032.gif" style="width:36px; height:15px" > is commutative if and only if both <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > and <m:omath><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image024.gif" style="width:8px; height:15px" > are commutative. Results and discussion Many basic properties of <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>A</m:r></m:e><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>♯</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image038.gif" style="width:12px; height:16px" >, some notions of amenability and some homological properties are extended to <m:omath><m:r>A</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>×</m:r></m:e><m:sub><m:r>θ</m:r></m:sub></m:ssub><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image032.gif" style="width:36px; height:15px" > by many authors. In particular, Ghaderi, Nasr-Isfahani and Nemati extended some results on (approximate) cyclic amenability of <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>A</m:r></m:e><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>♯</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image038.gif" style="width:12px; height:16px" >, obtained by Esslamzadeh and Shojaee, to <m:omath><m:r>A</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>×</m:r></m:e><m:sub><m:r>θ</m:r></m:sub></m:ssub><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image032.gif" style="width:36px; height:15px" >. They showed that if <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>A</m:r></m:e><m:sup><m:r>2</m:r></m:sup></m:ssup></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image048.gif" style="width:13px; height:15px" > is dense in <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > then the cyclic amenability <m:omath><m:r>A</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>×</m:r></m:e><m:sub><m:r>θ</m:r></m:sub></m:ssub><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image032.gif" style="width:36px; height:15px" > is equivalent to the cyclic amenability of both <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > and <m:omath><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image024.gif" style="width:8px; height:15px" >. In this paper, by characterizing of cyclic derivations on Lau product <m:omath><m:r>A</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>×</m:r></m:e><m:sub><m:r>θ</m:r></m:sub></m:ssub><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image032.gif" style="width:36px; height:15px" > and module extension <m:omath><m:r>A</m:r><m:r>&oplus;</m:r><m:r>X</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image018.gif" style="width:33px; height:15px" >, we present general necessary and sufficient conditions for those to be (approximate) cyclic amenable. This not only provides new results on (approximate) cyclic amenability of these type of Banach algebras but also improves some main results in this topic. In particular we show that, under mild condition, the cyclic amenability of <m:omath><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Tri</m:r><m:r>(</m:r><m:r>A</m:r><m:r>,</m:r><m:r>X</m:r><m:r>,</m:r><m:r>B</m:r><m:r>)</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image050.gif" style="width:58px; height:15px" > is equivalent to the cyclic amenability of the corner algebras <m:omath><m:r>A</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image002.gif" style="width:8px; height:15px" > and <m:omath><m:r>B</m:r></m:omath><img alt="" src="file:///C:UsersUser1AppDataLocalTempmsohtmlclip1�1clip_image024.gif" style="width:8px; height:15px" >. جبر باناخ, گسترش مدولی, حاصل ضرب لائو, میانگین‌پذیری دوری (تقریبی) Banach algebra, module extension, Lau product, (approximate) cyclic amenability 89 105 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1094-1&slc_lang=fa&sid=1 Mohammad Ramezanpour محمد رمضانپور ramezanpour@du.ac.ir 10031947532846005465 10031947532846005465 Yes Damghan University دانشگاه دامغان Mahdieh Alikahi مهدیه علی کاهی mahdiehalikahi91@gmail.com 10031947532846005466 10031947532846005466 No Damghan University دانشگاه دامغان