I wish to use the following finite-dimensional Banach spaces. although they do not need to be Banach algebras for my proposed application, a mild curiosity is aroused, because the multiplication proposed is continuous, and the spectrum and ideal structure is particularly simple which might make them a useful elementary teaching device. however the usual norm restriction on multiplication: $$\mid\mid xy\mid\mid \le \mid\mid x\mid\mid \; \mid\mid y\mid\mid $$ is not always valid. does this mean the spaces are not Banach algebras, and if so, what are the problems?
the Banach spaces $B_s$
for integer $s \gt 0$ let $B_s$ be the vector space of s-tuples of real numbers, with componentwise multiplication. define a norm for $b \in B_s$: $$ \mid \mid b \mid \mid = \frac1s \sum \mid b_k \mid $$ we define then positive cone $K_s^+$ as the set of elements $b$ for which all $b_k \gt 0$, and note that the multiplicative identity $1_s$, with all components equal to unity, satisfies: $$ \mid \mid 1_s \mid \mid = 1$$ the minimal ideals of $B_s$ are the obvious 1-dimensional subspaces generated by an orthogonal set of idempotents $e_k=(\delta_{jk})_{j=1\dots s}$. and the set of all ideals is isomorphic to the Boolean algebra on $s$ atoms.
the spectrum of an element of $B_s$ is simply the multiset of real numbers which occur in its $s$-tuple. a non-repeated value therefore corresponds to an eigenspace which is a minimal ideal. the intersection of all maximal ideals is the zero ideal, and the space is the direct sum of all its minimal ideals with: $$1_s = \sum e_k$$
the spectral radius of an element is equal to its norm in the $sup$ metric.
This is a long comment. It seems that the key difference happens in infinite dimensions and it's probably worth studying a specific example of this. Consider a toy example in finite dimensions. Let $B$ be the space of $n\times n$ matrices $A$ with the max norm: $\|A\|:=\max\{|a_{ij}|\}$. This is not a submultiplicitive norm. Of course, it's still a Banach Algebra in the loose sense that it's closed under addition and multiplication.
We can generalize this to infinite dimensions by taking the space of infinite matrices, and we define matrix multiplication as usual with $\{AB\}_{ij}=\sum_{k=1}^\infty a_{ik}b_{kj}$. Now we can try to single out the "bad" pairs of $A,B$ which are not submultiplictive under the max-norm above and study if you wish, their spectral properties.