Finite-dimensional C*-algebras and their (unital) *-homomorphisms are introduced. The main examples are introduced beginning with function spaces over finite sets to direct sums of matrix algebras.

Notes:
1. I forgot to mention that the involution * is conjugate-linear (instead of linear).
2. When I say a *-homomorphism preserves all the algebra structure, I mean it’s an algebra homomorphism (it’s linear and preserves products), it preserves *, and it preserves the unit. However, the norm need not be preserved. There are *-homomorphisms that can decrease the norm (but they cannot increase it!). Nevertheless, any injective *-homomorphism is an isometry (this is proved in Dixmier’s book on C*-algebras).
3. I said that 1 does not equal 0 in the definition of a C*-algebra, but you don’t want that restriction. Indeed, the set of functions on the empty set is a perfectly fine C*-algebra and corresponds to 0 (in fact, any algebra over the complex numbers where 0=1 is isomorphic this one). One thing to note about this is that there are NO (unital) homomorphisms from this one to a non-zero algebra. However, the 0 algebra is terminal as there is a unique homomorphism from any algebra into it.
4. Finally, I also forgot to write U* on the right-hand-side of the last example.

Reference: Attal’s book “Lectures in Quantum Noise Theory” (http://math.univ-lyon1.fr/~attal/chapters.html), specifically Lecture 6, reviews many of these things in detail and with greater generality.
Another reference: Chapter 2 of https://arxiv.org/abs/1708.00091

This is part of a series of lectures on special topics in linear algebra. It is assumed the viewer has taken (or is well into) a course in linear algebra. Some topics may also require additional background. Topics covered include linear regression and data analysis, ordinary linear differential equations, differential operators, function spaces, category-theoretic aspects of linear algebra, support vector machines (machine learning), Hamming’s error-correcting codes, stochastic maps and Markov chains, tensor products, finite-dimensional C*-algebras, algebraic probability theory, completely positive maps, aspects of quantum information theory, and more.

These videos were created during the 2019 Spring/Summer semester at the UConn CETL Lightboard Room.
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In our first lecture, we introduce the category of C*-algebras, within which our course will take place.
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