We prove that the triangular matrix algebras over any commutative unital $C^*$-algebra are Banach ${\ast}$-algebras and that the primitive ideals of these algebras and some of their Banach $ \ast $-subalgebras are all maximal. *******************************************************************************
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:In this paper we provide new examples of Banach $ \ast $-subalgebras of the matrix algebra $M_n(\mathscr{A}) $ over a commutative unital $C^*$-algebra $\mathscr{A}$.
📊
Temel Bulgular & Veri:For any involutive algebra, we define two involutions on the triangular matrix extensions.
💡
Sonuç & Katkı:We prove that the triangular matrix algebras over any commutative unital $C^*$-algebra are Banach ${\ast}$-algebras and that the primitive ideals of these algebras and some of their Banach $ \ast $-subalgebras are all maximal. *******************************************************************************
In this paper we provide new examples of Banach $ \ast $-subalgebras of the matrix algebra $M_n(\mathscr{A}) $ over a commutative unital $C^*$-algebra $\mathscr{A}$. For any involutive algebra, we define two involutions on the triangular matrix extensions. We prove that the triangular matrix algebras over any commutative unital $C^*$-algebra are Banach ${\ast}$-algebras and that the primitive ideals of these algebras and some of their Banach $ \ast $-subalgebras are all maximal. *******************************************************************************
Ahmadi, M., Moussavi, A. (2020). Involutive triangular matrix algebras. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 5. https://doi.org/10.15672/hujms.559837
O.M. Di Vincenzo, P. Koshlukov and R. La Scala, Involutions for upper triangular matrix algebras, Adv. in Appl. Math. 37, 541–568, 2006.
J. Dixmier, C*-Algebras, North-Holland, Amsterdam, 1977.
N. Jacobson, A topology for the set of primitive ideals in an arbitrary ring, Proc, Nat, Acad, Sci. U.S.A. 31, 333–338, 1945.
T.K. Lee and Y. Zhou, Armendariz and reduced rings, Comm. Algebra 32 (6), 2287– 2299, 2004.
G.J. Murphy, C*-Algebras and Operator Theory Academic Press, 1990.
T.W. Palmer, Banach Algebras and the General Theory of $\ast$-Algebras Volume I Al- gebras and Banach Algebras, Encyclopedia of Mathematics and its Applications, Vol. 1, 1994.
V. Paulsen, Completely Bounded Maps and Operator Algebras, vol. 78, Cambridge University Press, 2002.